## Abstract

Balanced Stable Marriage (BSM) is a central optimization version of the classic Stable Marriage (SM) problem. We study BSM from the viewpoint of Parameterized Complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, A(a), who a ranks strictly; we use p_{a}(b) to denote the position of b∈ A(a) in a’s preference list. The objective is to decide whether there exists a stable matching μ such that balance(formula presented). In SM, all stable matchings match the same set of agents, A* which can be computed in polynomial time. As balance (formula presented) for any stable matching μ, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to (formula presented). With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization (formula presented). The two extreme stable matchings are the man-optimal μ_{M} and the woman-optimal μ_{W}. Let (formula presented), and (formula presented). In this work, we prove that BSM parameterized by (formula presented) admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t.BSM parameterized by (formula presented) is W[1]-hard.

Original language | English |
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Title of host publication | Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings |

Editors | Zachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack |

Publisher | Springer Verlag |

Pages | 423-437 |

Number of pages | 15 |

ISBN (Print) | 9783030247652 |

DOIs | |

State | Published - 1 Jan 2019 |

Event | 16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada Duration: 5 Aug 2019 → 7 Aug 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11646 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Symposium on Algorithms and Data Structures, WADS 2019 |
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Country/Territory | Canada |

City | Edmonton |

Period | 5/08/19 → 7/08/19 |

## Keywords

- Balanced stable marriage
- Kernelization
- Parameterized complexity

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science